Sigma-delta (Σ-Δ) modulation is a widely used and thoroughly investigated technique for converting an analog signal into a high-frequency digital sequence. See, for example, “Oversampling Delta-Sigma Data Converters,” eds. J. C. Candy and G. C. Temes, IEEE Press, 1992, and “Delta-Sigma Data Converters,” eds. S. R. Northworthy, R. Schreier, G. C. Temes, IEEE Press, 1997, both of which are hereby incorporated herein by reference.
In Σ-Δ modulation, a low-resolution quantizer is incorporated within a feedback loop configuration in which the sampling frequency is much higher than the Nyquist frequency of the input signal (i.e., much higher than twice the maximum input frequency). In addition, the noise energy introduced in the quantizer is shaped towards higher frequencies according to a so called “noise transfer-function” NTF(z), and the signal passes the modulator more or less unchanged according to a so called “signal-transfer-function” STF(z).
FIG. 1(a) depicts a simple first order Σ-Δ modulator for a discrete time system having a subtraction stage 101, an accumulator 102 (including an integrator adder 103 and a delay line 104), and a one-bit quantizer 105. In normal operation, an input signal x(n) within the range [−a≦x(n)≦a] is converted to the binary output sequence ya(n) ε±a. The quantizer 105 produces a+1 for a positive input and a−1 for a negative input. The output from the quantizer 105 is fedback and subtracted from the input signal x(n) by the subtraction stage 101. Thus, the output of the subtraction stage 101 will represent the difference between the input signal x(n) and the quantizer output signal ya(n). As can be seen from FIG. 1(a), the output of the accumulator 102 represents the sum of its previous input and its previous output. Thus, depending on whether the output of the accumulator 102 is positive or negative, the one-bit quantizer 105 outputs a+1 or a−1 as appropriate.
As can be seen in FIG. 1(b), if the quantizer 105 is replaced by an adder 106 and a noise source 107 the basic relationship between the z-transforms of system input x(n), quantizer noise γa(n); and the two-level output sequence y(n) is:Ya(z)=z−1X(z)+(1−z−1)Γa(z),where index “a” denotes the amplitude of sequence ya(n), i.e., ya(n) ε±a. The signal transfer function and noise transfer function can be identified as STF(z)=z−1 and NTF(z)=(1−z−1), respectively.
For higher order modulators, the signal transfer function remains unchanged, and the noise transfer function becomes NTF(z)=(1−z−1)k, where k denotes the order of the modulator. The signal-transfer function STF(z)=z−1 means that the input signal is represented in the output sequence Ya(n), delayed by one sampling clock period. This transfer function does not contain any bandwidth limitations of the input signal. Any input signal x(n) within the range [−a +a] can be processed by the Σ-Δ modulator, including discontinuous signals with step-like transitions. For the modulator depicted in FIG. 1, this can easily be demonstrated, if it is regarded as a linear (non-adaptive) delta modulator, whose input is the accumulated input x(n). If the input is within the range [−a≦x(n)≦a], the magnitude of the maximum slope of the accumulated sequence x(n) is a/T (with T as sampling period). Thus, the delta modulator can always track its input, and so called “slope-overload conditions” cannot occur.
In most applications, this basic Σ-Δ feature is not exploited. In order to cut off the shaped out-of-band-quantization noise, the Σ-Δ output sequence ya(n) is low-pass filtered (usually by means of linear filters), thereby removing also the spectral components of x(n) outside the base band.